The scalar triple product is the determinant with the three vectors as columns. A cyclic permutation of three columns is even, while exchanging two columns reverses the determinant's sign. Consequently

(a \times b) \cdot c = a \cdot (b \times c) = - a \cdot (c \times b) .

Equivalently, expand the first expression as

ϵ_{ijk} a_{i} b_{j} c_{k}

and interchange the last two indices, using the antisymmetry of the Levi-Civita symbol. This argument includes linearly dependent vectors, for which both sides vanish.